<!DOCTYPE html>
<html lang="en-US">
<!--********************************************-->
<!--*       Generated from PreTeXt source      *-->
<!--*                                          *-->
<!--*         https://pretextbook.org          *-->
<!--*                                          *-->
<!--********************************************-->
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<meta name="robots" content="noindex, nofollow">
</head>
<body class="ignore-math">
<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>Then we intend to find a particular solution. Let</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_37.html ./knowl/eq3_39.html ./knowl/eq3_38.html ./knowl/eq3_41.html ./knowl/eq3_40.html">
\begin{equation}
Y=u_1 x^2+u_2 \frac{1}{x}.\tag{3.7.16}
\end{equation}
</div>
<p class="continuation">Then</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_37.html ./knowl/eq3_39.html ./knowl/eq3_38.html ./knowl/eq3_41.html ./knowl/eq3_40.html">
\begin{equation*}
Y^{\prime}=2 x u_1-x^{-2} u_2+\underline{u_1^{\prime} x^2+u_2^{\prime} x^{-1}}.
\end{equation*}
</div>
<p class="continuation">Choose</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_37.html ./knowl/eq3_39.html ./knowl/eq3_38.html ./knowl/eq3_41.html ./knowl/eq3_40.html">
\begin{equation}
u_1^{\prime} x^2+u_2^{\prime} x^{-1}=0.\tag{3.7.17}
\end{equation}
</div>
<p class="continuation">Then</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_37.html ./knowl/eq3_39.html ./knowl/eq3_38.html ./knowl/eq3_41.html ./knowl/eq3_40.html">
\begin{equation}
Y^{\prime \prime}=2 u_1+2 x^{-3} u_2+ 2 x u_1^{\prime}-x^{-2} u_2^{\prime}.\tag{3.7.18}
\end{equation}
</div>
<p class="continuation">Substituting (<a href="" class="xref" data-knowl="./knowl/eq3_37.html" title="Equation 3.7.16">(3.7.16)</a>) and (<a href="" class="xref" data-knowl="./knowl/eq3_39.html" title="Equation 3.7.18">(3.7.18)</a>) into (<a href="" class="xref" data-knowl="./knowl/eq3_38.html" title="Equation 3.7.13">(3.7.13)</a>),</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_37.html ./knowl/eq3_39.html ./knowl/eq3_38.html ./knowl/eq3_41.html ./knowl/eq3_40.html">
\begin{equation*}
x^2 (2 u_1+2 x^{-3} u_2+ 2 x u_1^{\prime}-x^{-2} u_2^{\prime})-2(u_1 x^2+u_2 x^{-1})=3 x^2-1,
\end{equation*}
</div>
<p class="continuation">which gives</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_37.html ./knowl/eq3_39.html ./knowl/eq3_38.html ./knowl/eq3_41.html ./knowl/eq3_40.html">
\begin{equation}
2 x^3 u_1^{\prime}-u_2^{\prime}=3 x^2-1.\tag{3.7.19}
\end{equation}
</div>
<p class="continuation">From (<a href="" class="xref" data-knowl="./knowl/eq3_41.html" title="Equation 3.7.17">(3.7.17)</a>) and (<a href="" class="xref" data-knowl="./knowl/eq3_40.html" title="Equation 3.7.19">(3.7.19)</a>), one has</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_37.html ./knowl/eq3_39.html ./knowl/eq3_38.html ./knowl/eq3_41.html ./knowl/eq3_40.html">
\begin{equation*}
u_1^{\prime}=\frac{1}{x}-\frac{1}{3 x^3}, \quad u_2^{\prime}=\frac{1}{3}-x^2.
\end{equation*}
</div>
<p class="continuation">Then the solutions to <span class="process-math">\(u_1\)</span> and <span class="process-math">\(u_2\)</span> are</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_37.html ./knowl/eq3_39.html ./knowl/eq3_38.html ./knowl/eq3_41.html ./knowl/eq3_40.html">
\begin{equation*}
u_1=\ln |x|+\frac{1}{6 x^2},\quad u_2=\frac{x}{3}-\frac{x^3}{3}.
\end{equation*}
</div>
<p class="continuation">Thus, the particular solution is</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_37.html ./knowl/eq3_39.html ./knowl/eq3_38.html ./knowl/eq3_41.html ./knowl/eq3_40.html">
\begin{equation*}
Y=x^2 \ln |x|+\frac{1}{6}+\frac{1}{3}-\frac{x^2}{3}=x^2 \ln |x|-\frac{x^2}{3}+\frac{1}{2}.
\end{equation*}
</div>
<p class="continuation">The general solution is</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq3_37.html ./knowl/eq3_39.html ./knowl/eq3_38.html ./knowl/eq3_41.html ./knowl/eq3_40.html">
\begin{equation*}
y=C_1 x^2+C_2 x^{-1}+x^2 \ln |x|-\frac{x^2}{3}+\frac{1}{2}=C_1 x^2 +C_2 x^{-1}+x^2 \ln |x|+\frac{1}{2}.
\end{equation*}
</div>
<span class="incontext"><a href="sec3_7.html#p-144" class="internal">in-context</a></span>
</body>
</html>
